Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T01:58:12.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of Quasi-Metrizable Bitopological Spaces

Part of: Spaces with richer structures

Published online by Cambridge University Press:  09 April 2009

T. G. Raghavan  and
I. L. Reilly
T. G. Raghavan
Affiliation:
Department of MathematicsUniversity of AucklandPrivate Bag, Auckland, New Zealand
I. L. Reilly
Affiliation:
Department of MathematicsUniversity of AucklandPrivate Bag, Auckland, New Zealand
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove that a pairwise Hausdorff bitopological space is quasi-metrizable if and only if for each point x ∈ X and for i, j = 1,2, i ≠ j, one can assign nbd bases { S(n, i; x) | n = 1, 2,… } such that (i) y ∉ S (n − 1, i; x) imples S(n, i; x) ∩ S (n, j; y) = φ, (ii) y ∈ S (n, i; x) implies S (n, i; y) ⊂ S(n − 1, i; x). We derive two further results from this.

Keywords

quasi-metricquasi-uniformitybitopological space

MSC classification

Secondary: 54E55: Bitopologies
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 271 - 274
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Kelly, J. C., ‘Bitopological spaces’, Proc. London Math. Soc. 13 (1963), 71–89.CrossRefGoogle Scholar
[2]Lane, E. P., ‘Bitopological spaces and quasi-uniform spaces’, Proc. London Math. Soc. 17 (1967), 241–256.CrossRefGoogle Scholar
[3]Murdeshwar, M. C. and Naimpally, S. A., Quasi-uniform topological spaces (Noordhoff, Groningen, 1966).Google Scholar
[4]Pareek, C. M., ‘Bitopological and quasi-metric spaces’, J. Univ. Kuwait Sci. 6 (1980), 1–7.Google Scholar
[5]Patty, C. W., ‘Bitopological spaces’, Duke Math. J. 34 (1967), 387–392.CrossRefGoogle Scholar
[6]Reilly, I. L., ‘Quasi-guage spaces’, J. London Math. Soc. 6 (1973), 481–487.CrossRefGoogle Scholar
[7]Romoguera, S., ‘Two characterizations of quasi-pseudo-metrizable bitopological spaces’, J. Austral. Math. Soc., 35 (1983), 327–333.CrossRefGoogle Scholar
[8]Romoguera, S., ‘On bitopological quasi-pseudometrization’, J. Austral. Math. Soc. 36 (1984), 126–129.CrossRefGoogle Scholar
[9]Salbany, S., ‘Quasi-metrization of bitopological spaces’, Arch. Math. 23 (1972), 299–316.CrossRefGoogle Scholar
[10]Stoltenberg, R., ‘Some properties of quasi-uniform spaces’, Proc. London Math. Soc. 17 (1967), 342–354.Google Scholar
[11]Wilson, W. A., ‘On quasi-metric spaces’, Amer. J. Math. 53 (1931), 675–684.CrossRefGoogle Scholar